Cayley form

A form in $ ( n + 1) ( N + 1) $ variables, where $ n = \mathop{\rm dim} X $, with $ X $ a closed algebraic subvariety of an $ N $- dimensional projective space $ \mathbf P ^ {N} $, which is determined by $ X $ uniquely up to a constant factor and itself uniquely determines $ X $. The exact definition is as follows. Let $ \widetilde{\mathbf P} {} ^ {N} $ be the $ N $- dimensional projective space of all hyperplanes in $ \mathbf P ^ {N} $, and let $ \Gamma $ be the subset of the variety

$$ {\widetilde{\mathbf P} } {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} \times X $$

( $ n + 1 $ factors $ \widetilde{\mathbf P} {} ^ {N} $) consisting of all $ ( n + 2) $- tuples $ ( \pi _ {1} \dots \pi _ {n + 1 } , x) $, where $ x \in X $ is a point in the intersection of the hyperplanes $ \pi _ {1} \dots \pi _ {n + 1 } $ and

$$ \phi : \widetilde{\mathbf P} {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} \times X \rightarrow \ {\widetilde{\mathbf P} } {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} $$

( $ n + 1 $ factors $ \widetilde{\mathbf P} {} ^ {N} $ each time) is the natural projection. Then $ \phi ( \Gamma ) $ is an irreducible subvariety of codimension 1 in

$$ {\widetilde{\mathbf P} } {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} $$

( $ n + 1 $ factors) and so $ \phi ( \Gamma ) $ is the zero variety for some form $ F _ {X} $ on

$$ {\widetilde{\mathbf P} } {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} $$

( $ n + 1 $ factors).

It may always be assumed that $ F _ {X} $ has no multiple factors, and this condition defines $ F _ {X} $ uniquely up to multiplication by a constant, given $ X $. Conversely, $ F _ {X} $ uniquely determines the set of all possible $ ( n + 1) $- tuples of hyperplanes in $ \mathbf P ^ {N} $ that intersect at points of $ X $, and so $ F _ {X} $ uniquely determines $ X $. The form $ F _ {X} $ is called the Cayley form of $ X $.

The Cayley form is also frequently known as the Chow form, or the associated form, of the variety $ X $. The idea of defining $ X $ by the complex of linear subspaces of dimension $ N - n - 1 $ in $ \mathbf P ^ {N} $ that intersect $ X $ goes back to A. Cayley [5], who applied it to the case $ n = 1 $, $ N = 3 $. The coefficients of the Cayley form are known as the Chow coordinates of the variety $ X $.

The Cayley form $ F _ {X} $ is homogeneous in each of the $ n + 1 $ coordinate systems of the space

$$ {\widetilde{\mathbf P} } {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} $$

( $ n + 1 $ factors) (the $ i $- th coordinate system is the coordinate system of the $ i $- th factor in this space). The form $ F _ {X} $ has the same degree of homogeneity with respect to each system; this common degree $ d $ is denoted by $ \mathop{\rm deg} X $ and is called the degree of the subvariety $ X $. It admits a geometrical interpretation: $ d $ is the maximum number of points of intersection of $ X $ with all possible $ ( N - n) $- dimensional linear spaces $ L $ in $ \mathbf P ^ {N} $ such that $ X \cap L $ is a finite set (i.e. $ d $ is the number of points in the intersection of $ X $ with a "general" $ ( N - n) $- dimensional linear subspace).

The set of all forms (up to multiplication by a non-zero constant) in $ n + 1 $ groups of variables consisting of $ N + 1 $ variables and having degree $ d $ in each group, forms a projective space $ \mathbf P ^ {\nu _ {N, n, d } } $ of dimension $ \nu _ {N, n, d } $. The Cayley form $ F _ {X} $ may be identified with a point of $ \mathbf P ^ {\nu _ {N, n, d } } $. The set $ C _ {N, n, d } $ of all points in $ \mathbf P ^ {\nu _ {N, n, d } } $ which are Cayley forms of $ n $- dimensional closed subvarieties of degree $ d $ in $ \mathbf P ^ {N} $ is a quasi-projective variety; it parametrizes the family of all such subvarieties in such a way that the family is algebraic relative to the parametrization. In general, $ C _ {N, n, d } $ need not be closed in $ \mathbf P ^ {\nu _ {N, n, d } } $.

The construction of the Cayley form carries over in a natural way to closed $ n $- dimensional cycles in $ \mathbf P ^ {N} $, i.e. to formal linear combinations $ m _ {1} X _ {1} + \dots + m _ {s} X _ {s} $ of closed $ n $- dimensional subvarieties $ X _ {1} \dots X _ {s} $ in $ \mathbf P ^ {N} $ with integer coefficients $ m _ {i} > 0 $. In fact, one puts

$$ \mathop{\rm deg} X = \ \sum _ {i = 1 } ^ { s } m _ {i} \mathop{\rm deg} X _ {i} $$

and $ F _ {X} = F _ {X _ {1} } ^ {m _ {1} } \dots F _ {X _ {s} } ^ {m _ {s} } $. The set $ {C _ {N, n, d } } bar $ of all Cayley forms of $ n $- dimensional cycles of degree $ d $ in $ \mathbf P ^ {N} $ is closed in $ \mathbf P ^ {\nu _ {N, n, d } } $.

The investigation of Cayley forms and the properties of the varieties $ C _ {N, n, d } $ and $ {C _ {N, n, d } } bar $ constitute an important aspect in the problem of classifying subvarieties and cycles in $ \mathbf P ^ {n} $. The first stage in this classification is to study the decomposition of $ {C _ {N, n, d } } bar $ into irreducible components. For example, if $ N = 3 $, $ n = 1 $, $ d = 2 $( curves of degree 2 in three-dimensional space), the variety $ {C _ {3, 1, 2 } } bar $ decomposes into two irreducible $ 8 $- dimensional components. The first corresponds to planar curves of the second order, the second to pairs of straight lines. The birational classification of the varieties $ C _ {N, n, d } $ is an important problem (in all known examples these varieties are rational).

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2]	W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 2 , Cambridge Univ. Press (1952) MR0048065 Zbl 0048.14502
[3]	P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1955) MR0072531
[4]	W.-L. Chow, B.L. van der Waerden, "Zur algebraische Geometrie IX" Math. Ann. , 113 (1937) pp. 692–704 Zbl 0016.04004
[5]	A. Cayley, "On a new analytical representation of curves in space" , Collected mathematical papers , 4 , Cambridge Univ. Press (1891) pp. 446–455 (Quart. J. Pure Appl. Math. 3 (1860), 225–236)

Comments

The varieties $ C _ {N,n,d} $ are special cases of Hilbert schemes (cf. Hilbert scheme).